Student

Spring term 2016

Bachelor’s Thesis, 15 ECTS

Civilekonomprogrammet

Crowdfunding and Economic Growth:

Potential Effects on Investment Efficiency

Johan Holmberg

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Acknowledgements:

Firstly, I want to thank my supervisor Kenneth Backlund for his support and guidance, two

key components for the realisation of this thesis. I would also like to thank the helpful people

at the institution who have helped in the process in various ways. Finally, would I like to

summarise the experience of writing this thesis with a quote:

“I do remember one thing. It took hours and hours, but by the time I was done with it, I was

so involved, I didn’t know what to think” – Adrian Belew

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Abstract:

Crowdfunding is an alternative form of finance that have emerged with the widespread adoption

of the internet. With the increasing utilization of crowdfunding, this thesis sets out to

theoretically investigate whether crowdfunding could affect economic growth. If the choice of

investment allocation mechanism could have any effect on the efficiency of investments made

in the economy. The results show that crowdfunding could have a potential effect on the leakage

of investments in the economy. The relative accuracy of the screening process and the

transaction costs coupled with the method used for conducting the investments could affect the

socially optimal proportion of investments conducted using crowdfunding.

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Table of content:

1 Introduction: .................................................................................................................................... 1

1.1 Background: .................................................................................................................................. 1

1.2 Factors of economic growth: Stating the importance of capital and innovation .......................... 2

1.3 Innovation: The discrepancy between holders of ideas and capital .............................................. 3

1.4 Leakage of investments: ................................................................................................................ 3

1.5 Problem and Purpose: ................................................................................................................... 5

1.6 Outline: .......................................................................................................................................... 5

2 Crowdfunding: ................................................................................................................................. 5

2.1 Explaining the phenomenon .......................................................................................................... 5

2.2 Debt Crowdfunding: ...................................................................................................................... 7

2.3 Equity Crowdfunding: ................................................................................................................... 7

2.4 Reward- and Donation Crowdfunding .......................................................................................... 7

2.5 Summary of Crowdfunding ............................................................................................................ 8

3 Theoretical foundation: ................................................................................................................... 8

3.1 The rationality of the crowd: ......................................................................................................... 8

3.2 Pre-market screening: ................................................................................................................. 10

3.3 Investments beyond money: ......................................................................................................... 11

3.4 Transaction costs for investments: .............................................................................................. 12

4. Analysis ......................................................................................................................................... 13

4.1 The Reference Model: ................................................................................................................. 13

4.2 The Social Optimum .................................................................................................................... 16

4.3 Optimal Proportions: .................................................................................................................. 19

5 Discussion: ..................................................................................................................................... 23

6 Reference list: ................................................................................................................................. iv

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1 Introduction:

1.1 Background:

Pod Point is a company formed in 2009 that provides products and services for electric motoring

(Pod Point, n.d.a). They sell chargers for companies, individuals and have started to build an

infrastructure for electric vehicles by creating a public network of chargers (Pod Point, n.d.a).

Seeking capital for product development Pod Point launched a campaign selling equity on the

crowdfunding website Seedrs in 2014 (Pod Point, 2014). The company sold 6.89% of its equity

raising £1.47 million (Pod Point, 2014). Following the previous success, and in order to raise

capital for a capital for an expansion, a second crowdfunding campaign was started. This time

they used the crowdfunding website Crowdcube, selling another 6.49% of its equity, this time

for £1,805 million (Pod Point, 2015) and mini-bonds for £373 thousand (Pod Point, n.d.b). By

running these public campaigns on different websites on the internet the company managed to

raise more than £3,6 million for their business.

Crowdfunding is a word used to describe various forms of fundraising, commonly via different

web based platforms (Ahlers et al., 2015, p. 955) and it’s an alternative form of financing where

consumers are able to provide capital for production (Kumar et al, 2015, p. 1). By creating new

opportunities for small companies or individual entrepreneurs to raise capital, crowdfunding

could potentially bridge the gap between the public and companies neglected by traditional

financial backers (Sigar, 2012, p. 481). This phenomenon is recognized as a possible way of

democratising the capital markets (Mollick & Robb, 2016, p. 73).

The body of published literature currently discussing crowdfunding is relatively limited but

growing. 477 peer reviewed articles on the subject were available via EBSCOhost as of 2016-

04-25. To my knowledge the literature discussing crowdfunding’s possibility to affect

economic growth is limited. This thesis aims to fill some of the gap by developing a theoretical

model based on the current literature and examine the possible growth effects of this additional

capital allocation mechanism. Previous literature has studied various aspects of crowdfunding.

Belleflamme et al. (2015) provides a description of crowdfunding, the design of crowdfunding

platforms and discusses underlying factors behind the decision to turn to crowdfunding. Mollick

and Robb (2016) discusses the possible democratization of the capital market as a result of

crowdfunding using the result of previous studies. Greenberg and Mollick (2014) examine

women’s historical disadvantage regarding raising capital for new ventures and crowdfunding

as an exception. Younkin and Kashkooli (2016) investigates crowdfunding as a solution to

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different problems including gatekeeping, also coordination, inexperience and patronage.

Mollick and Nanda (2014) examines the difference between the crowds and experts’ screening

of theatrical projects.

1.2 Factors of economic growth: Stating the importance of capital and innovation

Capital or producer goods has for a long time been acknowledged as an important factor of

production. In the classical economics by the likes of Smith and Ricardo capital is one of three

factors of production, the others being land and labour (Ricardo, 1817; Smith, 1776).

The influential Solow-Swan model of exogenous economic growth was written and published

independently by Robert Solow and Trevor Swan in 1956. This model has managed to remain

current to this day and includes several important insights. One of these was that the output of

a worker is partly a function of the amount of capital at his or her disposal (Solow, 1956, p. 69).

This view is agreed upon by many economists like James Tobin (1965) who also believed in a

positive effect of capital intensity on productivity (p. 673). According to this view, growth per

capita is partly a result of productivity changes due to a dynamic evolution of the capital stock.

The significance of capital accumulation for economic growth is debated but it’s difficult to

argue against the existence of positive productivity effects from capital goods. Both the strength

and somewhat the limitations of capital accumulation are visible in the Solow-Swan model.

The capital stock has an impact on productivity in the economy but this increased productivity

comes at the cost of decreased consumption in the present and rising costs of capital

depreciation. This conclusion of the capitals importance for production is fundamental for the

neoclassical and classical theory of economics.

A factor frequently addressed in the context of theories of economic growth is innovation and

technological efficiency. Joseph Schumpeter was one of the first economist to recognise

innovation as the driving factor of economic growth (Carlin & Soskie, 2006, p. 538). Philippe

Aghion and Peter Howitt later operationalised the concept of creative destruction, were new

technology replaces the old resulting in a turbulent process resulting in growth, in a model.

Economic growth is seen as the result of new innovation originating from the desire for

monopoly profits but the new technology render the old obsolete (Aghion et al., 2015, p. 94).

Generating new innovations demands labour which means that parts of the workforce have to

engage in R&D instead of production (Aghion et al., 2015, p. 94). Robert Solow concluded that

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roughly 87.7 % of the rise in productivity between 1909 and 1949 in the USA was due to

technical change increasing the productivity of the production factors (1957, p. 316). Later

statistical analysis conducted using additional explanatory variables estimated the impact of

technical change to be lower and explaining around a third of economic growth (Cameron,

1996, p. 2). The magnitude of the effect that increases in technological efficiency has on

economic growth is debated. However, many agree on the existence of positive productivity

effects from increasing technological efficiency.

1.3 Innovation: The discrepancy between holders of ideas and capital

One important insight from the Schumpeterian growth models is that new businesses or

innovations doesn’t spring out of thin air and create themselves, they are in some ways a

manifestation of the ideas of people. However, people with ideas are not always the same as

the people with the financial means necessary in order to make an attempt at realising them.

Inventors or entrepreneurs frequently have to search for external funds. This could mean that

they turn to bankers, venture capitalists or government officials in order to obtain capital.

Historically the privilege of deciding which ideas would receive monetary or institutional

support and subsequently a chance at commercialisation have been largely confined to these

small groups of experts (Mollick & Robb, 2016, p. 72). For example, venture capitalist are in a

position to select ideas deemed appropriate before the market can have a say and companies in

the seed stage denied finance has a disadvantage compared to companies granted funds. This is

partly due to the increased difficulty in creating business ties due to lack of financial means to

pay for goods and services (Ferrary & Granovetter, 2009, p. 346).

1.4 Leakage of investments:

Economic models acknowledge the importance of investment for the growth of production and

to an extent the consumption in the economy, whether the resources invested is monetary or

consists of labour. The investment entails a contemporary loss of consumption, either by

returning output to the production as investments or the opportunity cost in production resulting

from devoting parts of the workforce to R&D, in exchange for a higher future productivity.

The market economy is dynamic by nature. Over time new businesses enters and new products

are developed and introduced to the market. It is no secret that some of these ventures succeed

whilst other fails and that existing products or corporations may be forced out of the market.

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Sometimes new products are rejected by the market. Schumpeter distinguished between two

types of risks, technical failure of production and the risk of commercial failure (Schumpeter,

1934, p. 8). Between 1945 and 2004 then 19 empirical and peer reviewed studies placed the

new product failure rate in the range of 30-49% (Castellion & Markham, 2013, p. 978). Newly

founded corporations might also get rejected by the market. A study of Bavarian business

ventures found that 23.7% of newly founded businesses was closed after two years and 37.1%

after five (Brüderl et al., 1992, p. 235).

One might argue that this results in explicit and implicit costs from a socioeconomic

perspective. Resources invested in failed projects had most likely been more productive if

allocated for other uses. These inefficiencies in the allocation of investments on the market is

part of the business reality and rarely questioned in economic literature. One possible question

one could ask is if the choice of investment allocation mechanism could have any effect on this

inefficiency, if investments made by the consumers could outperform the investments executed

by the experts. Some argue that the deregulation of the equity crowdfunding market will result

in increased misplacement of investments. Isenberg (2012) argues that investments in early

stage equity is too difficult to standardise in a way which makes it comprehensible for

consumers while retaining functional and legal validity and that a crowd of consumers lack the

knowledge, time and funds necessary for conducting an adequate due diligence. Other studies

suggest that crowds could have the ability under certain conditions to match or even surpass the

judgements of experts (Budescu & Chen, 2014; Lorenz et al., 2011; Mollick & Nanda, 2015;

Surowiecki, 2004). Individual judgement can succumb to different kind of bias and aggregating

independent judgements of several individuals could under the right circumstances reduce this

bias (Budescu & Chen, 2014; Davis-Stober et al., 2014).

Beyond failure rates there is also complaints of an underrepresentation of women and minorities

in the venture capital market (Younkin & Kashkooli, 2016, p. 26). If there are obstacles for

women or minorities in the financial intermediary market, then crowdfunding could possibly

present a chance to circumvent these (Greenberg & Mollick, 2014, p. 35; Younkin & Kashkooli,

2016, p. 26).

A market with a perfect allocation of resources for product and business development could

seem improbable. However, the proportion of new product and business failure suggest that

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there might exist room for improvements. A reduction in the leakage of investments would

likely have positive effects on economic growth.

1.5 Problem and Purpose:

Could the possibility of using crowdfunding as an alternative process of allocating capital

increase efficiency of investments and thereby increase the growth rate of the capital stock and

technological development?

The purpose of this thesis is to analyse crowdfunding’s potential to decrease investment

inefficiencies and the possible welfare effects this could result in by its possible effect on capital

accumulation and technological development.

This is a theoretical study partly due to the decentralized nature of the crowdfunding phenomena

and partly due to the nature of the question investigated. The material used was found using

EBSCOhost, Google Scholar, Google and the library at Umeå university. This thesis is

theoretical in the sense that no empirical data was collected and analysed. The analysis will be

limited to financial return crowdfunding models were the crowd functions in a similar fashion

as traditional investors.

1.6 Outline:

A brief disposition of the thesis. Section 2 gives a description of crowdfunding. After that the

theoretical foundation of the model be depicted in section 3. In section 4 the reference model

will be presented followed by an analysis of the socially optimal proportions of investments

conducted using either crowdfunding or financial intermediaries. After that a rule for optimal

proportion will be presented. Lastly in section 5, there will be a discussion of the results and

some concluding remarks.

2 Crowdfunding:

2.1 Explaining the phenomenon

As previously discussed, the term crowdfunding is used to describe various forms of

fundraising, usually over the internet. The basic concept is that individuals with ideas and a

need of financial backing can present their ideas in a forum (for example websites like

Kickstarter, Indiegogo, Seedrs, Crowdcube and many others). Then the users of the site, or the

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crowd, are free to decide whether they want to fund the idea or not. Crowdfunding offers a route

to capital in addition to the previously existing ways of obtaining financial backing. It is

common that each individual backer makes a small contribution when dealing with

crowdfunding (Ahlers et al., 2015; Mollick & Kuppuswamy, 2014) and the volumes are instead

reached through a large number of contributors (Sigar, 2012, p. 478). Crowdfunding is a

growing phenomenon. According to Massolution’s annual crowdfunding industry reports the

amount of funds raised worldwide has increased from 1.5 billion USD in 2011 (Massolution,

2013) to 11.08 billion USD in 2014 (Massolution, 2015). A large number of websites are

participating in the market, crowdsourcing.org had as of 2016-04-06 a directory of 599

crowdfunding websites (Crowdsourcing, n.d).1

The underlying contracts between the investor and the recipient varies. Belleflamme et al.

distinguish between two broad categories of crowdfunding. One category consists of

crowdfunding based on investments and the other on crowdfunding based on donations and

rewards (Belleflamme et al., 2015, p. 12). Kirby and Worner define these as financial return-

and community crowdfunding (2014, p. 8). The first category consists of crowdfunding models

which includes monetary reward for the investor. In this category we find crowdfunding models

in where the crowd functions in a similar fashion to a traditional investor. The underlying

contract between supplier and recipient of funds can be based upon a loan contract, equity

shares in the venture or a fraction of future income (Belleflamme et al., 2015, p. 13). Donation-

and reward based crowdfunding are examples of models included in this category (Kirby &

Worner, 2014, p. 9).

Depending on the platform and the form of contracts currently used, the time when the recipient

receives the funds differ. “All or nothing” is a commonly applied type of contract clause which

require that a critical amount is obtained before any funds are distributed. Another type of

contract is the “keep it all” variety in this case the fundraiser receives the payments when they

arrive and keeps them whether any pre-determined goal is reached or not (Belleflamme, 2015,

p. 17).

1 This is to my knowledge the most complete register of sites, but it’s important to take possible coverage issues

into account.

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2.2 Debt Crowdfunding:

Debt crowdfunding is the leading model of crowdfunding in term of funding volume. In 2014

11.08 billion USD was raised via crowdfunding using debt contracts (Massolution, 2015).

These are principally the same as regular debt contracts but commonly in smaller

denominations than other common debt contracts like bank loans or corporate bonds. The

minimum required investment in a loan on the crowdfunding platforms Prosper and Lending

Club is $25 (LendingClub, n.d.; Prosper Marketplace, Inc, n.d.a). Both Prosper and

LendingClub function in a similar fashion to a credit rating agency and assign a credit grade to

each loan (LendingClub, n.d.; Prosper Marketplace, Inc, n.d.b) with the added element of

facilitating the interaction between lenders and debtors.

2.3 Equity Crowdfunding:

Equity crowdfunding entails a gathering of funds in exchange for equity stakes (Belleflamme

et al., 2015, p. 13). This form of financing constitutes of an open plead for funds over internet

where entrepreneurs hopes to catch the interest of a large group of investors (Ahlers et al., 2015,

p. 955). Equity crowdfunding became legally possible 2013 in the USA, but earlier US

legislation included limitations on individuals allowed to participate (Barnett, 2015).

Regulations on equity crowdfunding varies across countries (Gabison, 2014, p. 408). The rights

following the ownership of these stakes varies across platforms. Investors using the

crowdfunding platform Crowdcube receives direct ownership of the shares (Crowdcube, n.d.).

Whereas the platform Seedrs holds the investors shares as a nominee (Seedrs, n.d.). Other

platforms like WeFunder holds the securities and acts as a proxy holding the information and

voting rights2 (WeFunder, n.d.).

2.4 Reward- and Donation Crowdfunding

The reward and donation-based models for crowdfunding is a prominent form of crowdfunding

and constitutes a large portion of the overall crowdfunding market. During 2014 3.26 billion

USD were transferred between agents using these contracts (Belleflamme et al., 2015, p. 13).

With reward-based crowdfunding the investor receives a non-monetary return for the

investment. This return can consist of an article of the finished product and other additional

rewards depending on the amount invested (Hu et al., 2015, p. 332). This is the model used by

platforms such as Kickstarter and Indiegogo. Kickstarter is currently one of the largest

platforms for reward-based crowdfunding. According to themselves they have as of 2016-05-

2 If the start-up is using a WeFund which is the prominent form of fundraising on the site (WeFunder, n.d.).

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10 raised more than 2.36 billion and have had 10.8 million different backers since the launch in

2009 (Kickstarter, n.d.b). There are also crowdfunding models that are donation-based. In these

cases, the donor isn’t offered any reward for the donation. Examples of crowdfunding platforms

operating using this model could be Patreon which allows individuals to become patrons of

artists or bloggers etcetera giving a fixed donation for new creations (Patreon, 2013) or Razoo

which provides a platform for charity fundraising (Razoo, n.d).

2.5 Summary of Crowdfunding

One can conclude that crowdfunding is a phrase describing a set of different financial

interactions usually coordinated using the internet where a large number of individuals respond

to an open plea for financial support. It also has increased in popularity during recent years and

could possibly grow in the future as well. But with increasing volumes of capital distributed

using this mechanism there is also a rising importance of understanding the economic

implications of this phenomena.

3 Theoretical foundation of the model:

3.1 The rationality of the crowd:

Crowdfunding could potentially have a positive effect on the investment efficiency in the

economy. The accuracy of statistical aggregates of the opinions of a diverse crowd can

sometimes surpass the accuracy of individuals or even experts’ opinions when it comes to

forecasting and estimation tasks (Budescu & Chen, 2014, p. 267; Lorenz et al., 2011, p. 9020;

Surowiecki, 2005). Whether the wisdom of crowds is applicable to the funding of new start-

ups is a question up for debate. Group irrationality is well documented and when it comes to

financial backing of new business ventures then the crowd likely lacks the means necessary to

conduct a proper due diligence (Isenberg, 2012).

Francis Galton’s 1907 article Vox Populi is sometimes referred to as an early documentation of

the wisdom of crowds (Davis-Stober et al., 2014, p. 79). Galton investigated the accuracy of

the aggregated guesses made by 787 participants in a contest at a livestock show and found,

contrary to his expectations, that the aggregated estimations were surprisingly accurate. The

aggregated guess deviated less than one percent from the true value of the goal variable (Galton,

1907). A mathematical principle behind this phenomenon is the possibility of improving

statistical forecasts by combining different statistical methods using both simple and weighted

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averages. Combining forecasts can improve the accuracy and reduce the variability of

forecasting errors (Winkler & Makridakis, 1983, p. 157).3

Specific conditions need to be fulfilled before any crowd wisdom can be extracted. The crowd

has to be diverse and there has to be knowledge regarding the problem within the crowd,

incentives for the individuals to make their best guesses and these guesses have to be

independent (Simmons et al., 2009, p. 5). If the group of judges makes the same error then the

aggregated judgement will be biased (Simmons et al., 2009, p. 5). Ulrik Nash argues in his

article on systematic error in the wisdom of crowds that collective judgement is likely to be

skewed if the assessors’ judgements are more accurate than chance and the particular true value

is extreme compared to what is typical (2014, p. 11-12). Both articles suggest that the trust in

the crowd’s wisdom should be adjusted by the knowledge of the crowd (Nash, 2014; Simmons

et al., 2009).

Simmons et al. investigated the rationality of groups betting on American football over the

course of 17 weeks. The subjects in the study were divided into four different groups performing

prediction tasks under slightly different terms. Three groups predicted the outcome of a game

based on a point-spread which had been altered in the favour of the underdogs. One group was

uninformed of the alteration, one informed and the third bet on the point-spread before

predicting an exact point differential (Simmons et al., 2009, p. 14-15). The fourth group

predicted the exact point differential each game (Simmons et al., 2009, p. 15). The study found

a presence of systematic bias which meant that the crowd favoured the favourites even when

the odds were altered in favour of the underdogs (Simmons et al., 2009, p. 21-22). This was

prevalent both when the subjects were uninformed of the alteration of the odds and when the

subjects were informed of the alteration of the odds (Simmons et al., 2009, p. 22). Simmons et

al. concluded that the crowd was systematically biased and not wise (2009, p. 32). The fraction

of individuals beaten by the different crowds ranged from 0% for the crowd placing bets

informed of the altered odds to 57.8% for the group unconditionally predicting exact point

differentials (Simmons et al., 2009, p. 42). By reanalysing this data, Davis-Stober et al. got a

different result. They found that more than 50% of the individuals performed worse than the

crowd based on analysis using their expected loss model (Davis-Stober et al., 2014, p. 95).

3 They combined forecasts made using different methods, not by different people.

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Even if the crowd is uniformly and substantially biased there is evidence that the aggregated

judgement still can be superior to the judgement of a single high-performing non-biased

individual (Davis-Stober et al., 2014, p. 91). It seems like there’s currently no clear consensus

regarding what criteria needs to be fulfilled in order for the crowd to be considered wise. Davis-

Stober et al. defines crowd wisdom as when the squared expected error of the crowd is smaller

than that of the individual (2014, p. 83) and Simmons et al. arguably presumes a measurement

of crowd wisdom based upon the accuracy the crowds’ estimations relative to reality.

The wisdom of crowds-effect is diminishing when individual judgements aren’t independent,

social influence lead to higher confidence but lower accuracy of estimates and (Lorenz et al.,

2011, p. 9024) the threshold for herding behaviour, triggering convergence of estimates

reducing diversity without any positive effects on group accuracy, is low when it comes to

social influence (Lorenz et al., 2011, p. 9024).4

3.2 Pre-market screening:

Both financial intermediation and crowdfunding could provide a mechanism for pre-market

screening and selection of new concepts and ventures. Ferrary and Granovetter performed an

investigation of the role played by venture capital firms in Silicon Valley and stated that they

play a critical role in evaluating and selecting start-up ventures at the seed stage (2009, p. 345-

346). They also suggest that this pre-market selection saves resources if they are the ones best

qualified for this screening, but also point out that potentially valuable inventions could be

hindered from reaching the market by this process. Venture capitalists employ several different

routines for reducing the risk of an unfavourable outcome when investing, but still the most

likely primary outcome of a venture capital investment is failure (Lerner, 2002, p. F76).

Mollick and Nanda performed a study comparing expert- and crowd evaluation of theatre

projects. They found that the crowd and the experts generally agree when it comes to which

theatre are worthy of financial backing (2015, p. 17). In the most cases when the crowd and

experts disagree then the crowd is willing to back projects rejected by the experts (Mollick &

Nanda, 2015, p. 12). The study found no evidence that the projects backed only by the crowd

faired any worse than the projects backed by both (Mollick & Nanda, 2015, p. 16). The

4 All that was needed was to inform judges of the estimations of others, no additional social psychological

influences were allowed in the experiment (Lorenz et al., 2011, p. 9024).

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fundraising was online and experts were only asked to evaluate and state whether they would

back the project or not. Due to a low response rate and limited number of observations the

results should be interpreted carefully. One can’t say that this study provides any definitive

evidence of the crowds’ predictive capabilities matching that of the experts, but the article

provides some empirical support for the theory. Crowdfunding could also be a potential filler

of the financing gap left by decreasing bank lending after the recent financial crisis (Kirby &

Worner, 2014, p. 21) and it is possible that it could provide funding of small businesses which

otherwise would be unlikely to receive it (Borello et al., 2015, p. 17; Sigar, 2012, p. 481).

An effective pre-market selection should theoretically reduce the leakage of investments in the

economy. Relative accuracy of prediction and quality of the pre-market selection of the

crowdfunding crowd compared to the traditional gatekeepers or experts affects which capital

allocation mechanism would be most efficient. There is also concerns for the possibility of

fraud in crowdfunding settings which should be taken into account (Kirby & Worner, 2014, p.

5; Sigar, 2012, p. 481).

3.3 Investments beyond money:

Financial intermediaries can reduce moral hazard and adverse selection problems in the market

by thoroughly scrutinizing ventures before financing them and monitoring the entrepreneur

afterwards (Lerner, 2002, p. F75). This function can according to Lerner reduce some of the

capital constraints on the market. Financial intermediaries, like venture capitalist, can besides

from providing capital and screening provide access to networks by signalling the quality of

the start-up and provide valuable competence regarding managerial, technological, industrial

and legal issues (Ferrary & Granovetter, 2009, p. 348-349). These kinds of non-monetary assets

could mean the difference between making it through hardships or not for a new venture and

crowdfunded companies may not have access to them (Gobble, 2012, p. 6).

Crowdfunding comes with other non-monetary assets. This way of financing involves an early

test of the market potential for the concept. It also provides an opportunity for marketing

products and receiving feedback from the consumers (Sigar, 2012, p. 481). Using

crowdsourcing at the same time as crowdfunding provides a way of receiving support and

suggestions regarding product development (Sigar, 2012, p. 481).

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3.4 Transaction costs for investments:

Even if the allocation of capital would be perfect in the market there would be leakage in

investments due to transaction costs linked to the allocation mechanism. For example, financial

intermediaries have self-costs covered by generating income from maturity transformation or

fee generating activities. Costs like the interest rate spread and fees are a form of leakage

reducing the fraction of invested funds ultimately used for acquiring capital or financing

technological development. A possible approximation of this leakage in financial institutions

could be the net interest spread, the difference between the deposit and lending rates. According

to the Federal Reserve Bank of St. Louis this interest spread has averaged 3.49 percentage

points between the year 2000 and the third quarter of 2015 for commercial banks in the US (St.

Louis FED, n.d.). The present value of this cost over the duration of the loans could give an

approximation of the leakage from transaction costs for financial intermediaries when debt is

used.

There is no obvious equivalent to equity crowdfunding using financial intermediaries. The

underwriting commission of an IPO is around 5-7% (PWC, 2012, p. 8) this could give an

approximation of the transaction costs of selling private equity using financial intermediaries.

There are explicit costs linked to crowdfunding platforms as well, for example explicit costs

due to fees on the platform. These can take different shapes but a common method is to take a

fraction of the raised capital and in some “all or nothing” cases there’s also fees on funds

eventually returned. For example, the crowdfunding site Invesdor keeps transaction fees for

unsuccessful funding rounds (Invesdor, n.d.). The site Kickstarter takes a 5% fee of the funds,

not including payment processing fees, which range from 3-5% depending on the country and

average size of payments, from each successful fundraising project (Kickstarter, n.d.). Another

relatively large site Crowdcube has a similar fee structure on successful project but takes 6.5%

of received funds and has additional fixed fees for administration and corporate services

amounting to £2500, excluding VAT and payment processing fees (Crowdcube, n.d.b). There

are also other effects of including additional people to the project. Additional cost can arise

from this such as costs related to managing investor relations or administrative costs (Sigar,

2012, p. 482).

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4. Analysis

4.1 The Reference Model:

Let’s set up a dynamic model of capital accumulation and technological development including

crowdfunding. The goal of the model is to maximise the present value of all future consumption.

The objective function is written:

𝑀𝑎𝑥 ∫ 𝑢(𝐶𝑡)𝑒−𝑟𝑡𝑑𝑡∞

𝑡=0 (4.1.1)

where (u) represent utility, (C) consumption and (r) the discount rate. We will assume a positive

but declining marginal utility from consumption, or that 𝜕𝑢

𝜕𝐶> 0 and

𝜕2𝑢

𝜕𝐶2 < 0. In order to

consume we have to produce as well and output (Y) during any given time period is assumed

to be a function of capital stock (K), labour (L) and technological efficiency (A) at the time.

Technological efficiency is assumed to be a function of accumulated net investments in

technological development (Q). Writing in general form we get:

𝑌𝑡 = 𝑓(𝐾𝑡, 𝐿𝑡 , 𝐴𝑡(𝑄𝑡)) (4.1.2)

The first derivatives of output with respect to capital, labour and level of technological

development is assumed to be positive. To simplify the analysis, we keep population and

thereby labour constant and normalised to equal one. We will also normalise the relationship

between technological efficiency and accumulated net investments in technological

development. Investments will be allowed to be conducted in two ways in the model. Either via

financial intermediaries like banks or venture capital firms or directly from consumers via

crowdfunding. Other financing solutions exists, like investments using retained profits, but the

number of options will be restricted to two in order to simplify the analysis. The proportion of

total investments in capital goods conducted via financial intermediaries (𝜃𝐾) and the remaining

proportion of capital goods investments is assumed to be allocated by crowdfunding.

Representing a proportion, (𝜃𝐾) assumes values ranging from zero to one. There is an analogous

definition of the proportion of investments in technological development conducted using

financial intermediaries (𝜃𝑄).

A core concept in this model is the reduction of gross investments due to leakage in the form

of transaction costs and misplaced investments. The leakage is assumed to be a function of the

14

overall volume of investment and how the investments are conducted. There are two sources of

leakage in the model. One is the transaction costs (TC) linked to either path. This cost is

assumed to be proportional to the volume of funds. The other factor is the level of accuracy

between estimation and outcome, were invested funds in a project is viewed as a prediction of

success by the investor. The correlation between intermediaries’ prediction and the favourable

outcome is noted (Φ) and for the crowd (ψ). This notation and concept builds on the work by

Davis-Stober et al. (2014). These correlations are restricted to a value ranging between zero and

one. This model uses modified correlation between prediction and goal variable with the

correlation divided into two parts

Φ = 𝜑 + 𝛾𝐾𝐼(𝜃𝐾) 𝑤ℎ𝑒𝑟𝑒 𝜕𝛾𝐾𝐼

𝜕𝜃𝐾 ≤ 0 (4.1.3)

Ψ = 𝜓 + 𝛾𝐾𝐶𝐹(𝜃𝐾) 𝑤ℎ𝑒𝑟𝑒 𝜕𝛾𝐾𝐶𝐹

𝜕𝜃𝐾≥ 0 (4.1.4)

Φ = 𝜑 + 𝛾𝑄𝐼(𝜃𝑄) 𝑤ℎ𝑒𝑟𝑒 𝜕𝛾𝑄𝐼

𝜕𝜃𝑄 ≤ 0 (4.1.5)

Ψ = 𝜓 + 𝛾𝑄𝐶𝐹(𝜃𝑄) 𝑤ℎ𝑒𝑟𝑒 𝜕𝛾𝑄𝐶𝐹

𝜕𝜃𝑄≥ 0 (4.1.6)

Index K represent capital and index Q technological investments. The aggregated probability

of favourable outcomes for both investment pathways is modelled to depend on two factors.

The first factor noted (𝜓) for crowdfunding and (𝜑) for intermediaries, represent the ability to

pick the right project, in other words the statistical probability that any funded project will be

successful given that there’s no interaction between investor and recipient. The second part of

the functions is noted (γ) with four different index notations which represents the four different

possible paths for investments. (γ) represents increased likelihood of success due to active

intervention or interaction between investor and recipient of capital. This could for example be

the competitive advantage of having access to a venture capitalist’s business network as a result

of interaction with financial intermediaries or the possibility to crowdsource solution for

problems for the crowdfunded project. These are assumed to have a neutral or positive effect

on the likelihood of success, in other words γ ≥ 0, and that these advantages are assumed to be

decreasing or constant with increasing usage (see expressions 4.1.3–6). If we see the investors

efforts as a limited resource, an increasing number of different projects would lead to a lower

average per project.

15

(I) represents gross investments in capital goods and (Z) represent gross investments in

technological development, gross investments in the economy is determined as follows:

𝑌𝑡 − 𝐶𝑡 = 𝐼𝑡 + 𝑍𝑡 (4.1.7)

Gross investment in capital goods in a given time period is equal to what’s left of output (Y)

after consumption (C) and gross investments in technological development (Z). It’s possible to

see from 4.1.7 that investments in capital goods is partially a function of investments in

technological development. Every increment of investment in technological development (Z)

lead to an equally large reduction in investments in capital goods (I). Investment in capital

goods doesn’t give a one for one increase in capital stock due to two reasons, one is the

depreciation of the existing capital stock during that period (δK) and the other culprit is the

leakage of investment. The fraction of investments (I) conducted via intermediaries is

determined by 𝜃𝐾 and there are two types of leakages from this path. First we have leakage

from misplaced investments modelled as the fraction of net-investments after transaction costs

(TCI) lost due to an unfavourable project outcome. This cost decreases as accuracy of

intermediaries’ predictions (Φ(𝜃𝐾)) increases. There’s also transaction costs which are

independent of the project outcome. Transaction costs due to financial intermediaries is

assumed to be a fraction of net investments conducted using financial intermediaries. The

fraction of investments conducted via crowdfunding is the fraction not conducted via financial

intermediaries. The structure of the leakage is similar to that of the intermediaries, a fraction of

net investments is lost due to imperfect accuracy and there exists an inevitable transaction cost

which is independent of project outcome. By substituting (Y) using 4.1.2, capital accumulation

will be modelled:

𝜕𝐾

𝜕𝑡= 𝑓(𝐾𝑡, 1, 𝑄𝑡) − 𝐶 − 𝑍 − 𝛿𝐾 − 𝐼𝜃𝐾 ((1 − 𝑇𝐶𝐼)(1 − Φ(𝜃𝐾))+𝑇𝐶𝐼)

−𝐼(1 − 𝜃𝐾) ((1 − 𝑇𝐶𝐶𝐹)(1 − Ψ(𝜃𝐾)) + 𝑇𝐶𝐶𝐹) (4.1.8)

Accumulation of net investments in technological development will be modelled and

determined in a similar fashion.

𝜕𝑄

𝜕𝑡= 𝑍 − 𝑍𝜃𝑄 ((1 − 𝑇𝐶𝐼) (1 − Φ(𝜃𝑄)) +𝑇𝐶𝐼)

16

−𝑍(1 − 𝜃𝑄) ((1 − 𝑇𝐶𝐶𝐹) (1 − Ψ(𝜃𝑄)) + 𝑇𝐶𝐶𝐹) (4.1.9)

Investment in technological development (Z) have a leakage function similar to capital

investments.

4.2 The Social Optimum

The present value Hamiltonian function with an infinite-time horizon will be used to determine

the socially optimal course of action. The Hamiltonian function can be used to solve dynamic

constrained optimisation problems (Hoy et al., 1996, p. 843)

𝐻𝑡 = 𝑢(𝐶𝑡)𝑒−𝑟𝑡 + 𝜆1𝜕𝐾

𝜕𝑡+ 𝜆2

𝜕𝑄

𝜕𝑡 (4.2.1)

The Hamiltonian multipliers (noted as λ1 and λ2) represents the shadow price or the marginal

effect on the objective function from changing the constraints (Hoy et al., p. 853). In the

following case λ1 and λ2 represent the present marginal utility of increasing the constraint 𝜕𝐾

𝜕𝑡

and 𝜕𝑄

𝜕𝑡 by one unit respectively.

These are the first order conditions of the Hamiltonian function, 4.2.1:

𝜕𝐻

𝜕𝐶=

𝜕𝑢(𝐶𝑡)𝑒−𝑟𝑡

𝜕𝐶− 𝜆1 = 0 (4.2.2)

𝜕𝐻

𝜕𝑍= 𝜆1 (−1 + 𝜃𝐾 ((1 − 𝑇𝐶𝐼)(1 − Φ(𝜃𝐾)) + 𝑇𝐶𝐼) + (1 − 𝜃𝐾)((1 − 𝑇𝐶𝐶𝐹)(1 −

Ψ(𝜃𝐾)) + 𝑇𝐶𝐶𝐹)) + 𝜆2 (1 − 𝜃𝐴((1 − 𝑇𝐶𝐼)(1 − Φ(𝜃𝐴)) + 𝑇𝐶𝐼) − (1 − 𝜃𝐴)((1 −

𝑇𝐶𝐶𝐹)(1 − Ψ(𝜃𝐴)) + 𝑇𝐶𝐶𝐹)) = 0 (4.2.3)

17

𝜕𝐻

𝜕𝜃𝐾= 𝜆1 (−𝐼 (1 − 𝜑(1 − 𝑇𝐶𝐼) − 𝛾𝐾𝐼(𝜃𝐾)(1 − 𝑇𝐶𝐼) −

𝜕𝛾𝐾𝐼(𝜃𝐾)

𝜕𝜃𝐾𝜃𝐾(1 − 𝑇𝐶𝐼)) +

𝐼 (1 − 𝜓(1 − 𝑇𝐶𝐶𝐹) − 𝛾𝐾𝐶𝐹(𝜃𝐾)(1 − 𝑇𝐶𝐶𝐹) +𝜕𝛾𝐾𝐶𝐹(𝜃𝐾)

𝜕𝜃𝐾(1 − 𝜃𝐾)(1 − 𝑇𝐶𝐶𝐹))) = 0

(4.2.4)

𝜕𝐻

𝜕𝜃𝑄= 𝜆2 (−𝑍 (1 − 𝜑(1 − 𝑇𝐶𝐼) − 𝛾𝑄𝐼(𝜃𝑄)(1 − 𝑇𝐶𝐼) −

𝜕𝛾𝑄𝐼(𝜃𝑄)

𝜕𝜃𝑄𝜃𝑄(1 − 𝑇𝐶𝐼)) + 𝑍 (1 −

𝜓(1 − 𝑇𝐶𝐶𝐹) − 𝛾𝑄𝐶𝐹(𝜃𝑄)(1 − 𝑇𝐶𝐶𝐹) +𝜕𝛾𝑄𝐶𝐹(𝜃𝑄)

𝜕𝜃𝑄(1 − 𝜃𝑄)(1 − 𝑇𝐶𝐶𝐹))) = 0 (4.2.5)

𝑑𝜆1

𝑑𝑡= −

𝜕𝐻

𝜕𝐾= −𝜆1 (

𝜕𝑓(𝐾𝑡,1,𝑄𝑡)

𝜕𝐾− 𝛿) = 0 (4.2.6)

𝑑𝜆2

𝑑𝑡= −

𝜕𝐻

𝜕𝑄= −𝜆1 (

𝜕𝑓(𝐾𝑡,1,𝑄𝑡)

𝜕𝑄) = 0 (4.2.7)

From the first order condition 4.2.2 is it possible to see that in optimum:

𝜕𝑢(𝐶𝑡)𝑒−𝑟𝑡

𝜕𝐶= 𝜆1

We see that in optimum the marginal utility of consumption should be equal to the shadow price

of investment. Since the shadow price emulates the increase of utility from a relaxed constraint

this can be interpreted as the marginal change in utility from increasing either consumption or

investment should be equally large in optimum.

From 4.2.3 we see that in optimum the marginal benefit in terms of utility of net investments in

capital goods should be the same as that of net investments in technological development:

𝜆1 (1 − 𝜃𝐾((1 − 𝑇𝐶𝐼)(1 − Φ(𝜃𝐾)) + 𝑇𝐶𝐼) − (1 − 𝜃𝐾)((1 − 𝑇𝐶𝐶𝐹)(1 − Ψ(𝜃𝐾)) +

𝑇𝐶𝐶𝐹)) = 𝜆2 (1 − 𝜃𝐴((1 − 𝑇𝐶𝐼)(1 − Φ(𝜃𝐴)) + 𝑇𝐶𝐼) − (1 − 𝜃𝐴)((1 − 𝑇𝐶𝐶𝐹)(1 −

Ψ(𝜃𝐴)) + 𝑇𝐶𝐶𝐹))

From 4.2.4 and 4.2.5 we see that in optimum:

18

𝐼 (1 − 𝜑(1 − 𝑇𝐶𝐼) − 𝛾𝐾𝐼(𝜃𝐾)(1 − 𝑇𝐶𝐼) +𝜕𝛾𝐾𝐼(𝜃𝐾)

𝜕𝜃𝐾𝜃𝐾(1 − 𝑇𝐶𝐼)) =

𝐼 (1 − 𝜓(1 − 𝑇𝐶𝐶𝐹) − 𝛾𝐾𝐶𝐹(𝜃𝐾)(1 − 𝑇𝐶𝐶𝐹) −𝜕𝛾𝐾𝐶𝐹(𝜃𝐾)

𝜕𝜃𝐾(1 − 𝜃𝐾)(1 − 𝑇𝐶𝐶𝐹))

and

𝑍 (1 − 𝜑(1 − 𝑇𝐶𝐼) − 𝛾𝑄𝐼(𝜃𝑄)(1 − 𝑇𝐶𝐼) +𝜕𝛾𝑄𝐼(𝜃𝑄)

𝜕𝜃𝑄𝜃𝑄(1 − 𝑇𝐶𝐼)) =

𝑍 (1 − 𝜓(1 − 𝑇𝐶𝐶𝐹) − 𝛾𝑄𝐶𝐹(𝜃𝑄)(1 − 𝑇𝐶𝐶𝐹) −𝜕𝛾𝑄𝐶𝐹(𝜃𝑄)

𝜕𝜃𝑄(1 − 𝜃𝑄)(1 − 𝑇𝐶𝐶𝐹))

It is possible to see that when the proportions are optimal the marginal leakage from investing

via financial intermediaries should be equal to the marginal leakage of investing through

crowdfunding. We also see that in the case that

𝜕𝛾𝐾𝐶𝐹(𝜃𝐾)

𝜕𝜃𝐾=

𝜕𝛾𝐾𝐼(𝜃𝐾)

𝜕𝜃𝐾= 0

and

𝜕𝛾𝑄𝐶𝐹(𝜃𝑄)

𝜕𝜃𝑄=

𝜕𝛾𝑄𝐼(𝜃𝑄)

𝜕𝜃𝑄= 0

then 4.2.4 and 4.2.5 simplifies to

𝐼(1 − Φ(1 − 𝑇𝐶𝐼)) = 𝐼(1 − Ψ(1 − 𝑇𝐶𝐶𝐹))

and

𝑍(1 − Φ(1 − 𝑇𝐶𝐼)) = 𝑍(1 − Ψ(1 − 𝑇𝐶𝐶𝐹))

Theta is no longer a decision variable in need of marginal adjustments, either we get the case

when one of the alternatives has strictly less leakage than the other. The optimal action would

then be to exclusively chose that alternative. We could also get a situation where none of the

alternatives are better than the other, if this is the case then marginally adjusting theta wouldn’t

affect the outcome. This means that we get a corner solution except for the case when the

investment leakage is equally large for both intermediary financing and crowdfunding, in this

case we have no unique optimum solution.

19

From the first order optimum conditions 4.2.6 and 4.2.7 we do get the following optimum

condition. Here we see that in optimum the marginal effect on output of increasing investments

in technology should be equal to the marginal effect of increasing investment in capital goods

reduced by the rising cost of capital depreciation. The consumable or investable surplus in

output from marginally investing in technology should be equal to that of marginally investing

in capital goods.

(𝜕𝑓(𝐾𝑡,1,𝑄𝑡)

𝜕𝑄) = (

𝜕𝑓(𝐾𝑡,1,𝑄𝑡)

𝜕𝐾− 𝛿)

4.3 Optimal Proportions:

To be able to derive a rule for optimal proportion of investments conducted via financial

intermediaries then we have to specify the expressions describing the likelihood of a favourable

outcome. In order to simplify the analysis then the expressions will only consist of two parts

and be modelled as follows:

Φ(𝜃𝐾) = �̂�𝐾 − 𝑚𝐾𝐼𝜃𝐾𝛼𝐾 where �̂�𝐾 = Φ(𝜃𝐾=0) = 𝜑 + 𝛾𝐾𝐼

𝑀𝑎𝑥 (4.3.1)

Ψ(𝜃𝐾) = �̂�𝐾 + 𝑚𝐾𝐶𝐹𝜃𝐾𝛽𝐾 where �̂�𝐾 = Ψ(𝜃𝐾=0) = 𝜓 + 𝛾𝐾𝐶𝐹

𝑀𝑖𝑛 (4.3.2)

Φ(𝜃𝑄) = �̂�𝑄 − 𝑚𝑄𝐼𝜃𝑄

𝛼𝑄 where �̂�𝑄 = Φ(𝜃𝑄=0) = 𝜑 + 𝛾𝑄𝐼

𝑀𝑎𝑥 (4.3.3)

Ψ(𝜃𝐾) = �̂�𝑄 + 𝑚𝑄𝐶𝐹𝜃𝑄

𝛽𝑄 where �̂�𝑄 = Ψ(𝜃𝑄=0) = 𝜓 + 𝛾𝑄𝐶𝐹

𝑀𝑖𝑛 (4.3.4)

The accented and indexed phi and psi represents the aggregated probability of favourable

outcomes when all investments are conducted using crowdfunding, or when theta is equal to

zero. As the proportion of funds conducted via using either path the value of the active

intervention of the financiers is assumed to decrease. As the proportions of investments

increases the capability of these financiers is diluted over a larger number of projects, leading

to a marginal decrease of the positive effects for any given project. The decrease in the effect

of active intervention for financial intermediary investments and increase for crowdfunding

investments as the proportion of investments conducted using financial intermediaries changes

20

from zero to one is represented by (𝑚𝑥𝐶𝐹) and (𝑚𝑥𝐼). Inserting these expressions into the

Hamiltonian function we get new first order conditions.

𝜕𝐻

𝜕𝜃𝐾= 𝜆1 (−𝐼(1 − �̂�𝐾(1 − 𝑇𝐶𝐼) + 𝑚𝐾𝐼(1 − 𝑇𝐶𝐼)𝜃𝐾

𝛼𝐾(1 + 𝛼𝐾)) + 𝐼(1 − �̂�𝐾(1 − 𝑇𝐶𝐶𝐹) −

𝑚𝐾𝐶𝐹(1 − 𝑇𝐶𝐶𝐹)𝜃𝐾𝛽𝐾 + 𝛽𝐾𝑚𝐾𝐶𝐹(1 − 𝑇𝐶𝐶𝐹)(1 − 𝜃𝐾)𝜃𝐾

𝛽𝐾−1)) = 0 (4.3.5)

and

𝜕𝐻

𝜕𝜃𝑄= 𝜆2 (−𝑍 (1 − �̂�𝑄

(1 − 𝑇𝐶𝐼) + 𝑚𝑄𝐼(1 − 𝑇𝐶𝐼)𝜃𝑄𝛼𝑄(1 + 𝛼𝑄)) + 𝑍 (1 − �̂�𝑄

(1 − 𝑇𝐶𝐶𝐹) −

𝑚𝑄𝐶𝐹(1 − 𝑇𝐶𝐶𝐹)𝜃𝑄

𝛽𝑄 + 𝛽𝑄𝑚𝑄𝐶𝐹(1 − 𝑇𝐶𝐶𝐹)(1 − 𝜃𝑄)𝜃𝑄

𝛽𝑄−1)) = 0 (4.3.6)

From 4.3.5 and 4.3.6 we see that in optimum:

�̂�𝐾(1 − 𝑇𝐶𝐼) − 𝑚𝐾𝐼(1 − 𝑇𝐶𝐼)𝜃𝐾𝛼𝐾(1 + 𝛼𝐾) =

�̂�𝐾(1 − 𝑇𝐶𝐶𝐹) + 𝑚𝐾𝐶𝐹(1 − 𝑇𝐶𝐶𝐹)𝜃𝐾𝛽𝐾 − 𝛽𝐾𝑚𝐾𝐶𝐹(1 − 𝑇𝐶𝐶𝐹)(1 − 𝜃𝐾)𝜃𝐾

𝛽𝐾−1

respectively

�̂�𝑄(1 − 𝑇𝐶𝐼) − 𝑚𝑄𝐼(1 − 𝑇𝐶𝐼)𝜃𝑄𝛼𝑄(1 + 𝛼𝑄) =

�̂�𝑄(1 − 𝑇𝐶𝐶𝐹) + 𝑚𝑄𝐶𝐹(1 − 𝑇𝐶𝐶𝐹)𝜃𝑄

𝛽𝑄 − 𝛽𝑄𝑚𝑄𝐶𝐹(1 − 𝑇𝐶𝐶𝐹)(1 − 𝜃𝑄)𝜃𝑄

𝛽𝑄−1

We can rearrange expression 4.3.5 and simplify in order to obtain the implicit function:

𝐹 = �̂�𝐾(1 − 𝑇𝐶𝐼) − �̂�𝐾(1 − 𝑇𝐶𝐶𝐹) − 𝑚𝐾𝐼(1 − 𝑇𝐶𝐼)𝜃𝐾𝛼𝐾(1 + 𝛼𝐾)

−𝑚𝐾𝐶𝐹(1 − 𝑇𝐶𝐶𝐹)𝜃𝐾𝛽𝐾 + 𝛽𝐾𝑚𝐾𝐶𝐹(1 − 𝑇𝐶𝐶𝐹)(1 − 𝜃𝐾)𝜃𝐾

𝛽𝐾−1= 0 (4.3.7)

From the implicit function theorem, we know that:

𝜕𝜃𝐾

𝜕𝑋= −

𝜕𝐹𝜕𝑋⁄

𝜕𝐹𝜕𝜃𝐾

⁄ (4.3.8)

We can use this to analyse the effect on optimal proportion by changes in the other variables

in the function. 4.3.7 has the following derivatives:

21

𝜕𝐹

𝜕𝜃𝐾= −𝑚𝐾𝐼(1 − 𝑇𝐶𝐼)𝜃𝐾

𝛼𝐾−1𝛼𝐾(1 + 𝛼𝐾) − 𝑚𝐾𝐶𝐹(1 − 𝑇𝐶𝐶𝐹)𝜃𝐾𝛽𝐾−1

𝛽𝐾

+𝑚𝐾𝐶𝐹(1 − 𝑇𝐶𝐶𝐹)𝜃𝐾𝛽𝐾−2

𝛽𝐾(𝛽𝐾 − 1)

(4.3.9)

𝜕𝐹

𝜕�̂�𝐾= (1 − 𝑇𝐶𝐼) (4.3.10)

𝜕𝐹

𝜕�̂�𝐾= −(1 − 𝑇𝐶𝐶𝐹) (4.3.11)

𝜕𝐹

𝜕𝑇𝐶𝐼= −�̂�𝐾 + 𝑚𝐾𝐼𝜃𝐾

𝛼𝐾(1 + 𝛼𝐾) (4.3.12)

𝜕𝐹

𝜕𝑇𝐶𝐶𝐹= �̂�𝐾 + 𝑚𝐾𝐶𝐹𝜃𝐾

𝛽𝐾 − 𝛽𝐾𝑚𝐾𝐶𝐹(1 − 𝜃𝐾)𝜃𝐾𝛽𝐾−1

(4.3.13)

𝜕𝐹

𝜕𝑚𝐾𝐼 = −(1 − 𝑇𝐶𝐼)𝜃𝐾

𝛼𝐾(1 + 𝛼𝐾) (4.3.14)

𝜕𝐹

𝜕𝑚𝐾𝐶𝐹 = −(1 − 𝑇𝐶𝐶𝐹)𝜃𝐾

𝛽𝐾−1(𝜃𝐾 − 𝛽𝐾(1 − 𝜃𝐾)) (4.3.15)

𝜕𝐹

𝜕𝛼𝐾= −𝑚𝐾𝐼𝜃𝐾

𝛼𝐾(1 − 𝑇𝐶𝐼)(ln(𝜃𝐾)𝛼𝐾 + ln(𝜃𝐾) + 1) (4.3.16)

𝜕𝐹

𝜕𝛽𝐾 = 𝑚𝐾𝐶𝐹𝜃𝐾

𝛽𝐾−1 (1 − 𝑇𝐶𝐶𝐹)((ln(𝜃𝐾) 𝛽𝐾 − 𝜃𝐾ln(𝜃𝐾) + 1) (4.3.17)

We could rearrange 4.3.9 to get:

𝜕𝐹

𝜕𝜃𝐾= −

𝑚𝐾𝐼(1−𝑇𝐶𝐼)(𝛼𝐾+𝛼𝐾2 )𝜃𝐾

𝛼𝐾+1+𝜃𝐾𝛽

(𝑚𝐾𝐶𝐹(1−𝑇𝐶𝐶𝐹)𝛽𝜃𝐾+𝑚𝐾𝐶𝐹(1−𝑇𝐶𝐶𝐹)(𝛽−𝛽2))

𝜃𝐾2

The following analysis will be conducted under the assumption that 4.3.9 is negative.

Now can we analyse how optimal 𝜃𝐾 changes when the other variables in 4.3.7 changes.

From 4.3.10 we see that an increase of �̂�𝐾 would lead to an increase in optimal 𝜃𝐾. This could

be interpreted as if the accuracy of the financial intermediaries at its highest increases, ceteris

paribus, then the optimal proportion of investments conducted using financial intermediaries’

22

increases as well. In 4.3.11 a reduction of optimal 𝜃𝐾 from an increase in �̂�𝐾 is observable. If

the minimum accuracy of the crowdfunding crowd would increase, all else equal, then the

proportion of investments conducted using crowdfunding increases in optimum. If one of the

investment pathways become more efficient as an effect of a rising accuracy, all else equal,

then we should increase the usage of that pathway. From 4.3.12 we can see that if transaction

costs for financial intermediaries’ increases then we get two effects on optimal 𝜃𝐾 working in

opposite directions. Firstly, if the value of maximum accuracy were to decrease due to a rising

inevitable cost when using this investment mechanism, as observable in the first term, a

reduction of the optimal proportion of investments conducted via financial intermediaries

would occur. Secondly, then the importance of decreasing likelihood of a favourable outcome

due to increased usage of the investment pathway is also reduced due to increasing inevitable

leakage costs. This effect works in the opposite direction increasing optimal 𝜃𝐾. The

transaction costs of crowdfunding have a slightly different effect on optimal theta as seen in

4.3.13. The first two parts of the derivative shows that as transaction costs for crowdfunding

increases then the value of the current accuracy of investments conducted using crowdfunding

decreases, so optimal 𝜃𝐾 increases. The third part shows a similar result, the increase in

accuracy as a result of reduced usage loses its value as transaction costs increases. We would

expect that the accuracy at any 𝜃𝐾 is larger than the expected change in accuracy with increased

usage for both investment pathways. This would mean that if transaction costs for one of the

alternative increases, ceteris paribus, then the efficiency of that alternative decreases. This

would lead to a decreased usage of that investment pathway in the social optimum.

To specify exactly how optimal 𝜃𝐾 changes when a variable changes without specifying the

other terms would not be possible. The direction of the derivatives comes from the derivative

of the implicit function, 4.3.8 and 4.3.9. Since the derivative is divides by and then multiplied

with a negative these cancel out each other.

In order to derive a condition for optimal theta then it’s necessary to make assumption regarding

the exponentials. We could assume that we have a linear reduction of the benefits from active

interventions from increased usage, that 𝛼𝐾 = 𝛽𝐾 = 1. Then we could write 4.3.5 as:

�̂�𝐾(1 − 𝑇𝐶𝐼) − �̂�𝐾(1 − 𝑇𝐶𝐶𝐹)+ 𝑚𝐾𝐶𝐹(1 − 𝑇𝐶𝐶𝐹) =

2𝑚𝐾𝐼(1 − 𝑇𝐶𝐼)𝜃𝐾1 + 2𝑚𝐾𝐶𝐹(1 − 𝑇𝐶𝐶𝐹)𝜃𝐾

1

23

We would get the following condition for optimal 𝜃𝐾:

𝜃𝐾∗ =

�̂�𝐾(1−𝑇𝐶𝐼)−�̂�𝐾(1−𝑇𝐶𝐶𝐹)+ 𝑚𝐾𝐶𝐹(1−𝑇𝐶𝐶𝐹)

2(𝑚𝐾𝐼(1−𝑇𝐶𝐼)+𝑚𝐾𝐶𝐹(1−𝑇𝐶𝐶𝐹)) (4.3.18)

The proportion of investments conducted using financial intermediaries is in 4.3.18 determined

by the probability of a favourable outcome for financial intermediaries at its highest and the

likelihood of a favourable outcome for investments conducted using crowdfunding at its lowest

both with their respective transaction costs and the rates of which these probabilities changes

with usage taken into account.

5 Discussion:

The ability to evaluate and pick projects worthy of financial backing is crucial in order for

crowdfunding to be an efficient mechanism for allocating investments in the economy. To what

extent a crowd is wise is an ongoing debate. There is a theoretical possibility of the crowd’s

judgements being inferior, equal or even superior to the judgements of experts. More research

has to be done in order to make any definitive statements and it is likely to vary depending on

the branch of industry or nature of the project seeking funds.

I would argue for the possibility of a different relative accuracy of expert and crowd evaluation

depending on the nature of the evaluated project. For example, entertainment is a field were

consumer preferences should be considered a critical factor for commercial success. From a

market demand point of view, it could be said that a concepts ability to convince a large number

of potential consumers to part with money could give a stronger signal regarding the market

potential of the concept than the ability to convince one or a few financial gatekeepers. Mollick

and Nanda (2015) found some similarities in the outcome of a screening of theatre projects by

the crowdfunding crowd and experts. In these industries then it is plausible that Ψ ≥ Φ could

be true. For other branches then the general knowledge regarding different subjects could be a

hindrance for crowdfunding. It might be difficult for a crowd to evaluate the merits of projects

dealing with developments of new medicine or developing catalytic converter for cars without

any or limited knowledge of the subject. We would expect that Ψ < Φ in these cases. The

relative accuracy of the crowd in comparison to the experts is likely to vary depending of the

nature of the problem. Sometimes you could argue for a greater importance of knowledge about

24

the demand-side of the market and for these projects it is possible that Ψ > Φ, or that the crowds

predictions could be more accurate than that of the experts. Other projects might be more

demanding of knowledge regarding the supply side of the market. We could expect that Ψ <

Φ, that the predictions of experts would be more accurate than that of the crowdfunding crowd

in these cases. A uniform optimal proportion for the overall market would therefore seem

unlikely.

It’s important not to overstate the importance of the financers for a project beyond their

monetary contribution. The change in the likelihood of favourable outcomes due to a change

the quantity of investments conducted using any given mechanism is likely to be modest, that

the difference between (γ) at its highest and its lowest, or (m), is small. We would then observe

many situations were one alternative would be strictly better than the other.

It is possible to question whether the websites used for crowdfunding provides the right

circumstances for the wisdom of crowds to occur. Lorenz et al. (2011) shows that very little

social influence is needed to offset the wisdom of crowds. As little as presenting the guesses of

others has the potential of producing herding behaviour and reducing the quality of collective

judgement. Popular sites feature elements like progress bars which shows the progression of

investment toward funding goals, highlights different projects or shows how many people that

have backed any given project. This mean that the independence of the judgements could be

compromised reducing the accuracy of the collective judgements.

One problem with crowdfunding right now from a socioeconomic point of view is the high

transaction costs which reduces the potential positive effects of crowdfunding. If we have

transaction costs in the area of 8-10% of invested volume for crowdfunding, due to costs for

using the platforms and payment processing fees, and a 5-7% transaction cost for financial

intermediaries, then a significant superiority in the accuracy of crowd is needed to offset this

difference and make it more socioeconomically beneficial than financial intermediaries. An

interest rate spread for banks at 3.5% means that the present value of the transaction cost will

vary depending on the length of the debt contract and the discount rate (r). For debt contracts

with short times to maturity, around 2.8 years or less, then crowdfunding would lead to higher

transaction costs than financial intermediaries. For longer contracts then this would be

dependent on the discount rate, if the discount rate is positive then the present value of the

transaction cost for debt contract with longer times to maturity would be lower for the financial

25

intermediaries. These transaction costs pose a problem for crowdfunding from an economics

point of view, but it is possible that the costs could be reduced in the future as a result of

technical advancements or economics of scale effects. As it is now, then the relatively higher

transaction cost for crowdfunding in comparison to financial intermediaries could explain why

crowdfunding constitutes only a small fraction of the financial market.

Crowdfunding is an exciting new form of finance and it has grown significantly over the last

couple of years. As the quantity of investments conducted using crowdfunding increases then

importance of understanding the phenomena and its socioeconomic implications has increased

as well.

iv

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